A294032 Triangle read by rows, T(n, k) = Pochhammer(3, k)*Stirling2(3 + n, 3 + k) for n >= 0 and 0 <= k <= n.

1, 6, 3, 25, 30, 12, 90, 195, 180, 60, 301, 1050, 1680, 1260, 360, 966, 5103, 12600, 15960, 10080, 2520, 3025, 23310, 83412, 158760, 166320, 90720, 20160, 9330, 102315, 510300, 1369620, 2116800, 1890000, 907200, 181440

Offset: 0

Author

Peter Luschny, Oct 22 2017

Keywords

• nonn
• tabl

Examples

			Triangle starts:
[0]    1
[1]    6,      3
[2]   25,     30,     12
[3]   90,    195,    180,      60
[4]  301,   1050,   1680,    1260,     360
[5]  966,   5103,  12600,   15960,   10080,    2520
[6] 3025,  23310,  83412,  158760,  166320,   90720,  20160
[7] 9330, 102315, 510300, 1369620, 2116800, 1890000, 907200, 181440
		

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Crossrefs

T(n, 0) = A000392(n+3), T(n, n) = A001710(n+2).

Row sums A002051(n+3), alternating row sums A000225(n+1).

Cf. A028246 (m=1), A053440 (m=2), this seq. (m=3), A293617 (hub).

Programs

Maple

A294032 := (n, k) -> pochhammer(3, k)*Stirling2(n + 3, k + 3):
seq(seq(A294032(n, k), k=0..n), n=0..7);
T := (n, k) -> A293617(3, n, k): seq(seq(T(n, k), k=0..n), n=0..7);

Mathematica

Table[Pochhammer[3, k] StirlingS2[3 + n, 3 + k], {n, 0, 7}, {k, 0, n}] // Flatten (* Michael De Vlieger, Oct 22 2017 *)

PARI

for(n=0,10, for(k=0,n, print1((k+2)!*stirling(n+3,k+3,2)/2, ", "))) \\ G. C. Greubel, Nov 19 2017

Formula

E.g.f.: (1/2)*exp(x)*(2*y + 9*exp(2*x) + y^2+1-11*exp(3*x)*y + 15*y^2*exp(2*x) - 7*y^2*exp(x) - 13*y^2*exp(3*x) + 4*exp(4*x)*y^2 - 8*exp(x) + 24*y*exp(2*x) - 15*y*exp(x))/(1 - y*(exp(x) - 1))^3.

T(n, k) = A293617(3, n, k).